Remark 3.2 Hopf cylinders over curves in \(H^2\) 
[resp. \(S^2_1\) ] are surfaces in \(G\)  which are
invariant under \(K\) -action [resp. \(A\mathbf {Z}_2\) -action].
Surfaces considered in this section are
invariant under \(N\) -action.
Thus all the surfaces investigated in preceding section
and present section are invariant under
1-dimensional closed subgroup of the isometry
group \(G\times K\) .
In [1]}, Figueroa, Mercuri and
Pedrosa classified all
constant mean curvature surfaces
in the Heisenberg group which are invariant
under 1-dimensional closed subgroups
of the isometry group.
Some results in
[1]} are independently
obtained in
[3]}.
Recently S. D. Pauls studied minimal surfaces
in the Heisenberg group with
Carnot-Carathéodory metric [4]}

